{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# $\\lambda$ variable for prolate ellipsoids"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here, we follow the reasoning presented by [Webster (1904)](#webster-dynamics) for analyzing the ellipsoidal coordinate  $\\lambda$ describing a prolate ellipsoid.\n",
    "\n",
    "Let's consider an ellipsoid with semi-axes $a$, $b$, $c$ oriented along the $x$-, $y$-, and $z$-axis, respectively, where $a > b = c > 0$. This ellipsoid is defined by the following equation:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='eq1'></a>\n",
    "$$\n",
    "\\frac{x^{2}}{a^{2}} + \\frac{y^{2} + z^{2}}{b^{2}} = 1 \\: . \\tag{1}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A quadric surface which is confocal with the ellipsoid defined in [equation 1](#eq1) can be described as follows:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='eq2'></a>\n",
    "\n",
    "$$\n",
    "\\frac{x^{2}}{a^{2} + \\rho} + \\frac{y^{2} + z^{2}}{b^{2} + \\rho}= 1 \\: , \\tag{2}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "where $\\rho$ is a real number. We know that [equation 2](#eq2) represents an ellipsoid for $\\rho$ satisfying the condition"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='eq3'></a>\n",
    "\n",
    "$$\n",
    "\\rho + b^{2} > 0 \\: . \\tag{3}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Given $a$, $b$, and a $\\rho$ satisfying [equation 3](#eq3), we may use [equation 2](#eq2) for determining a set of points $(x, y, z)$ lying on the surface of an ellipsoid confocal with that one defined in [equation 1](#eq1). Now, consider the problem of determining the ellipsoid which is confocal with that one defined in [equation 1](#eq1) and pass through a particular point $(x, y, z)$. This problem consists in determining the real number $\\rho$ that, given $a$, $b$, $x$, $y$, and $z$, satisfies the [equation 2](#eq2)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "By rearranging [equation 2](#eq2), we obtain the following quadratic equation for $\\rho$:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "f(\\rho) = (a^{2} + \\rho)(b^{2} + \\rho) - (b^{2} + \\rho) \\, x^{2}\n",
    "- (a^{2} + \\rho) \\, (y^{2} + z^{2}) \\: .\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This equation shows that:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\rho = \\begin{cases}\n",
    "d \\to \\infty \\: &, \\quad f(\\rho) > 0 \\\\\n",
    "-b^{2} \\: &, \\quad f(\\rho) < 0 \\\\\n",
    "-a^{2} \\: &, \\quad f(\\rho) > 0\n",
    "\\end{cases} \\: .\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "By rearanging this equation, we obtain a simpler one given by:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='eq4'></a>\n",
    "\n",
    "$$\n",
    "f(\\rho) = p_{2} \\, \\rho^{2} + p_{1} \\, \\rho + p_{0} \\: , \\tag{4}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "where"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='eq5'></a>\n",
    "\n",
    "$$\n",
    "p_{2} = 1 \\: , \\tag{5}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='eq6'></a>\n",
    "\n",
    "$$\n",
    "p_{1} =  a^{2} + b^{2} - x^{2} - y^{2} - z^{2} \\tag{6}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='eq7'></a>\n",
    "\n",
    "$$\n",
    "p_{0} =  a^{2} \\, b^{2} - b^{2} \\, x^{2} - a^{2} \\, y^{2} - a^{2} \\, z^{2} \\: . \\tag{7}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that a particular $\\rho$ satisfying [equation 2](#eq2) results in $f(\\rho) = 0$ ([equation 4](#eq4)).\n",
    "\n",
    "In order to illustrate the parameter $\\rho$, consider the constants $a$, $b$, $x$, $y$, and $z$ given in the cell below:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "a = 20.\n",
    "b = 13.\n",
    "x = 21.\n",
    "y = 23.\n",
    "z = 30."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "By using these constants, we calculate the coefficients $p_{2}$ ([equation 5](#eq5)), $p_{1}$ ([equation 6](#eq6)) and $p_{0}$ ([equation 7](#eq7)) as follows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "p2 = 1.\n",
    "p1 = a**2 + b**2 - (x**2) - (y**2) - (z**2)\n",
    "p0 = (a*b)**2 - (b*x)**2 - (a*y)**2 - (a*z)**2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the sequence, we define a set of values for the variable $\\rho$ in an interval $\\left[ \\rho_{min} \\, , \\rho_{max} \\right]$ and evaluate the quadratic equation $f(\\rho)$ ([equation 4](#eq4))."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "rho_min = -a**2 - 100.\n",
    "rho_max = -b**2 + 2000.\n",
    "rho = np.linspace(rho_min, rho_max, 100)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "f = p2*(rho**2) + p1*rho + p0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, the cell below shows the quadratic equation $f(\\rho)$ ([equation 4](#eq4)) evaluated in the range $\\left[ \\rho_{min} \\, , \\rho_{max} \\right]$ defined above."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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zJ3PnzgWgcuXKtGrV6hT3ECmEsrK8Zjsry+8k+Ra6ssu///1vrc9dwmTlZJG2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\nuOIwAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0xa69c198>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ymin = np.min(f) - 0.1*(np.max(f) - np.min(f))\n",
    "ymax = np.max(f) + 0.1*(np.max(f) - np.min(f))\n",
    "\n",
    "plt.close('all')\n",
    "plt.figure(figsize=(10,4))\n",
    "\n",
    "plt.subplot(1,2,1)\n",
    "plt.plot([rho_min, rho_max], [0., 0.], 'k-')\n",
    "plt.plot([-a**2, -a**2], [ymin, ymax], 'r--', label = '$-a^{2}$')\n",
    "plt.plot([-b**2, -b**2], [ymin, ymax], 'g--', label = '$-b^{2}$')\n",
    "plt.plot(rho, f, 'k-', linewidth=2.)\n",
    "plt.xlim(rho_min, rho_max)\n",
    "plt.ylim(ymin, ymax)\n",
    "plt.legend(loc = 'best')\n",
    "plt.xlabel('$\\\\rho$', fontsize = 20)\n",
    "plt.ylabel('$f(\\\\rho)$', fontsize = 20)\n",
    "\n",
    "plt.subplot(1,2,2)\n",
    "plt.plot([rho_min, rho_max], [0., 0.], 'k-')\n",
    "plt.plot([-a**2, -a**2], [ymin, ymax], 'r--', label = '$-a^{2}$')\n",
    "plt.plot([-b**2, -b**2], [ymin, ymax], 'g--', label = '$-b^{2}$')\n",
    "plt.plot(rho, f, 'k-', linewidth=2.)\n",
    "plt.xlim(rho_min, 0.)\n",
    "plt.ylim(-0.5*10**6, 10**5)\n",
    "plt.legend(loc = 'best')\n",
    "plt.xlabel('$\\\\rho$', fontsize = 20)\n",
    "#plt.ylabel('$f(\\\\rho)$', fontsize = 20)\n",
    "\n",
    "plt.tight_layout()\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Remember that we are interested in a $\\rho$ satisfying [equation 3](#eq3). Consequently, according to the figures shown above, we are interested in the largest root $\\lambda$ of the quadratic equation $f(\\rho)$ ([equation 4](#eq4))."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The largest root $\\lambda$ of $f(\\rho)$ ([equation 4](#eq4)) is given by:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='eq8'></a>\n",
    "\n",
    "$$\n",
    "\\lambda = \\frac{-p_{1} + \\sqrt{\\Delta}}{2} \\: , \\tag{8}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "where"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='eq9'></a>\n",
    "\n",
    "$$\n",
    "\\Delta = p_{1}^{2} - 4 \\, p_{0} \\: . \\tag{9}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The cells below use the equations [8](#eq8) and [9](#eq9) to compute the root $\\lambda$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "delta = p1**2 - 4.*p2*p0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "lamb = (-p1 + np.sqrt(delta))/(2.*p2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "lambda = 1651.33927\n"
     ]
    }
   ],
   "source": [
    "print 'lambda = %.5f' % lamb"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "By substituing $\\lambda$ in [equation 4](#eq4), we can verify that it is a root of $f(\\rho)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "f_lamb = p2*(lamb**2) + p1*lamb + p0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "f(lambda) = -0.00000\n"
     ]
    }
   ],
   "source": [
    "print 'f(lambda) = %.5f' % f_lamb"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### References"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='webster-dynamics'></a>\n",
    "\n",
    "* Webster, A. G. 1904. The Dynamics of Particles and of Rigid, Elastic and Fluid Bodies. Universidade de Michigan."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "anaconda-cloud": {},
  "kernelspec": {
   "display_name": "Python [default]",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.10"
  }
 },
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